Singing and playing at the same time is like learning to drum. Anyone can do it it just needs some coordination and practice.
Ok i'm definetly no expert, particularly because i can't sing good to start with but i have a few ways of learning.
Start off with an easy song, lets say knocking on heavens door.
Ok skip doing it in straight 8th's because you can do that already. Instead ad some rhythm to it, don't do something to hard. Challenge your self by making the strums out of time with the vocals if you get what i mean.
Now what you need to is know exactly when to sing each note, exactly what note falls on each beat etc. This is real important, a lot of the problems i found form singing and playing at the same time was getting lost on where i should sing etc because i was getting thrown off by my rhythm.
First start playing the rhythm, G D C whatever. Do it really really really slow. Begin tapping your foot for each quater beat too. Now while your playing that, just hum the vocal lines slowly over the guitar, while tapping your foot making sure you're keeping everything in time. By doing it at this speed you'll find it a bit easier. And make sure you start your notes on the proper beats etc. A normal habit is to sing a note as you pick something. Practice singing notes on off beats from your strumming.
I couldn't sing and play KOHD at the same time, then i used this method and realized i was singing notes at the wrong time. Look at all the places where a note starts and see what you should be doing on guitar. Take it slow and it'll come.
Start next with single note riffs. Nirvana's Come as you are is pretty tough, try that for a real challenge. Just follow these steps.
Oh yeah and im no expert, i have a terrible voice but thats just my advice. Plenty of people have different methods i dont know the correct one.
Oh yeah and i find it much much much easier to play the piano and sing at the same time, weird.